= Solution
A <primitive Dirichlet character> modulo $q$ is a <Dirichlet character> which is not induced from a character of a proper divisor of $q$. To determine the inducing <primitive Dirichlet character>, use the <Chinese remainder theorem> to decompose
$$
(\mathbb Z/q\mathbb Z)^\times\cong\prod_{p^k\parallel q}(\mathbb Z/p^k\mathbb Z)^\times.
$$
On each factor choose the least exponent $c_p\in\{0,\ldots,k\}$ through whose reduction the restricted character factors. Exponent zero means the trivial unit group modulo one. Put $q_0=\prod_pp^{c_p}$ and define $\chi_0$ on the units modulo $q_0$ by these descended factors, extending by zero off the units. Every reduction of unit groups is <surjective>, so the descended character is unique. Its local exponents cannot be decreased, hence it is primitive. The original character is $\chi(n)=\chi_0(n)$ when $(n,q)=1$, and zero otherwise.
Any other inducing modulus must have exponent at least $c_p$ at every <prime>, by restriction to the corresponding local factor. Therefore $q_0$ is the unique minimal modulus, the <conductor of a Dirichlet character>, and $\chi_0$ is the unique <primitive Dirichlet character> inducing $\chi$. The argument also explains why removing extra <prime> factors can change values at <integers> that were nonunits for $q$.
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